Results 1 to 2 of 2

Math Help - densities -- confusing topic for me

  1. #1
    Newbie
    Joined
    Nov 2007
    Posts
    16

    densities -- confusing topic for me

    Random variables X and Y have joint density:
    f_x,y(x,y) = c(y^(2)-x^(2))e^(-y) when -y<x<y, y>0
    f_x,y(x,y) = 0 otherwise
    Here, c is a constant.

    I have shown that Y has a gamma density, and have found that c=1/8.

    Now, how do I find the density of Y^(3)?
    Any help is greatly appreciated! Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
    Posts
    397
    we can find the density of Y by integrating the joint density with respect to x
    thus getting
    fy(Y) = (1/6)*(y^3)*e^(-y)

    now we are interested in finding the pdf of z = g(y) = y^3, g(y) is a monotonic function of y thus the resulting density function is given by:

    fz(Z) = abs(1 / g'(g-1(z)) )*fy(g-1(z))

    where g-1 is the inverse of g in our case g-1(z) = z^(1/3), thus we get:

    fz(Z) = [(1/3)*Z^(-2/3)]*(1/6)*Z*e^(-Z^(1/3)) =
    [(1/18)*Z^(1/3)]*e^(-Z^(1/3)) for 0 <= Z <= inf
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Joint Densities
    Posted in the Statistics Forum
    Replies: 1
    Last Post: April 1st 2010, 02:58 AM
  2. Conditional Densities Question
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: December 2nd 2009, 10:08 PM
  3. Distribution function and Densities
    Posted in the Advanced Statistics Forum
    Replies: 8
    Last Post: October 14th 2009, 05:53 AM
  4. probability: conditioning and densities
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: December 4th 2007, 05:01 AM
  5. joint densities
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 14th 2007, 03:20 AM

Search Tags


/mathhelpforum @mathhelpforum